Existence and approximation of solutions for Fredholm equations of the first kind

Dan Butnariu; Benzion Shklyar

Existence and approximation of solutions for Fredholm equations of the first kind

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The Cimmino algorithm is an iterative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.

Original language	English
Title of host publication	WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.
Pages	181-185
Number of pages	5
State	Published - 2007
Event	11th World Multi-Conference on Systemics, Cybernetics and Informatics, WMSCI 2007, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Orlando, FL, United States Duration: 8 Jul 2007 → 11 Jul 2007

Publication series

Name	WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.
Volume	4

Conference

Conference	11th World Multi-Conference on Systemics, Cybernetics and Informatics, WMSCI 2007, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007
Country/Territory	United States
City	Orlando, FL
Period	8/07/07 → 11/07/07

Keywords

Almost common point
Asymptotic center of a sequence
Cimmino type algorithm
Discrete linear moment problem
Eigenvalue of a linear operator
Fredholm equation of the first kind
Gram matrix
Projection method
Spectrum of a linear operator

ASJC Scopus subject areas

Artificial Intelligence
Information Systems

Cite this

Butnariu, D., & Shklyar, B. (2007). Existence and approximation of solutions for Fredholm equations of the first kind. In WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc. (pp. 181-185). (WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.; Vol. 4).

Butnariu, Dan ; Shklyar, Benzion. / Existence and approximation of solutions for Fredholm equations of the first kind. WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.. 2007. pp. 181-185 (WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.).

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title = "Existence and approximation of solutions for Fredholm equations of the first kind",

abstract = "The Cimmino algorithm is an iterative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.",

keywords = "Almost common point, Asymptotic center of a sequence, Cimmino type algorithm, Discrete linear moment problem, Eigenvalue of a linear operator, Fredholm equation of the first kind, Gram matrix, Projection method, Spectrum of a linear operator",

author = "Dan Butnariu and Benzion Shklyar",

year = "2007",

language = "English",

isbn = "1934272183",

series = "WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.",

pages = "181--185",

booktitle = "WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.",

note = "11th World Multi-Conference on Systemics, Cybernetics and Informatics, WMSCI 2007, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 ; Conference date: 08-07-2007 Through 11-07-2007",

}

Butnariu, D & Shklyar, B 2007, Existence and approximation of solutions for Fredholm equations of the first kind. in WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.. WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc., vol. 4, pp. 181-185, 11th World Multi-Conference on Systemics, Cybernetics and Informatics, WMSCI 2007, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007, Orlando, FL, United States, 8/07/07.

Existence and approximation of solutions for Fredholm equations of the first kind. / Butnariu, Dan; Shklyar, Benzion.
WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.. 2007. p. 181-185 (WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.; Vol. 4).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Shklyar, Benzion

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N2 - The Cimmino algorithm is an iterative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.

AB - The Cimmino algorithm is an iterative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.

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KW - Projection method

KW - Spectrum of a linear operator

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SN - 9781934272183

T3 - WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.

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T2 - 11th World Multi-Conference on Systemics, Cybernetics and Informatics, WMSCI 2007, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007

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Butnariu D, Shklyar B. Existence and approximation of solutions for Fredholm equations of the first kind. In WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.. 2007. p. 181-185. (WMSCI 2007 - The 11th World Multi-Conference on Systemics, Cybernetics and Informatics, Jointly with the 13th International Conference on Information Systems Analysis and Synthesis, ISAS 2007 - Proc.).

Existence and approximation of solutions for Fredholm equations of the first kind

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

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